Comparing upper broadcast domination and boundary independence broadcast numbers of graphs

Mynhardt, Kieka; Neilson, Linda

doi:10.22108/toc.2023.127904.1836

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	Comparing upper broadcast domination and boundary independence broadcast numbers of graphs
Transactions on Combinatorics
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 07 بهمن 1401 اصل مقاله (628.61 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2023.127904.1836
نویسندگان
Kieka Mynhardt^* ¹؛ Linda Neilson²
¹Department of Mathematics and Statistics, University of Victoria, P. O.Box 3800, Victoria, Canada
²Department of Adult Basic Education, Vancouver Island University Nanaimo,Canada
چکیده
A broadcast on a nontrivial connected graph $G=(V,E)$ is a function $f:V\rightarrow\{0, 1,\dots,d\}$, where $d=\operatorname{diam}(G)$, such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$. The weight of $f$ is $\sigma(f)={\textstyle\sum_{v\in V}} f(v)$. A vertex $u$ hears $f$ from $v$ if $f(v)>0$ and $d(u,v)\leq f(v)$. A broadcast $f$ is dominating if every vertex of $G$ hears $f$. The upper broadcast domination number of $G$ is $\Gamma_{b}(G)=\max\left\{ \sigma(f):f\text{ is a minimal dominating broadcast of }G\right\}.$ A broadcast $f$ is boundary independent if, for any vertex $w$ that hears $f$ from vertices $v_{1},\ldots,v_{k},\ k\geq2$, the distance $d(w,v_{i})=f(v_{i})$ for each $i$. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number $\alpha_{\operatorname{bn}}(G)$. We compare $\alpha_{\operatorname{bn}}$ to $\Gamma_{b}$, showing that neither is an upper bound for the other. We show that the differences $\Gamma _{b}-\alpha_{\operatorname{bn}}$ and $\alpha_{\operatorname{bn}}-\Gamma_{b}$ are unbounded, the ratio $\alpha_{\operatorname{bn}}/\Gamma_{b}$ is bounded for all graphs, and $\Gamma_{b}/\alpha_{\operatorname{bn}}$ is bounded for bipartite graphs but unbounded in general.
کلیدواژه‌ها
broadcast domination؛ broadcast independence؛ hearing independent broadcast؛ boundary independent broadcast

مراجع
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Comparing upper broadcast domination and boundary independence broadcast numbers of graphs